Dear Luisma,
We know from the evidence of place and river names that Basque i-bai "river" was *bai or *pai in Iberian (the additional anterior i- is found only in Basque).
The word also appears inscribed on Iberian libation bottles in the form baigar "river flame".
In Britain, Basque and Iberian b generally became w.
Let's look at some examples.
*wai "river" is found in numerous river names, some of which comprise this element alone, while others are compounds.
There are the Rivers Wey in Surrey and Dorset (Waie 675 and Wai- 939 respectively).
And the Rivers Wye in Herefordshire and South-West Wales, and in Derbyshire (Guoy 800 and Wey 1235 respectively).
The Medway in Kent was Meduuuaeian 764 (with dative n). The first element here is probably connected to Basque bede/beda "portal, entrance hall", and may indicate that the river was believed to carry the souls of the dead to the next world.
The Wylie in Wiltshire was Wileo 688, and may have been known to the Romans as Velox (a form corrupted by association with a Latin word for "rapid"). This probably represents an earlier *wai-loi "muddy river", and as such would be identical to Baelo (Rio de Valle, Andalucia).
The Welland in the East Midlands was Weolud 921. This may represent and earlier *wai-lodi "swollen river".
The Weaver in Cheshire was Weever 1133. This may have been *wai-ware "calm river".
The Wiske in North Yorkshire was Wisca 1100-15. This could have been *wai-iska "oak river" (a form frequently translated by the Iron Age Celtic conquerors to Derventios - modern Derwent).
Now, how do I translate *isca as "oak tree"? Well, I connect it with Basque ezki "linden tree". This may once have meant "oak" because ezkur (literally "linden nut") is actually "acorn".
This stem was borrowed by the Indo-Europeans who entered Europe at the end of the Bronze Age and is now reflected in such words as English ash and Latin aesculus "holm oak".
The River Wyre in Lancashire was Wir 1184, a late form. This may be the same as the lost river names found in Worcester (Uueogorna civitas 692), Wyre Forest (Weogorena leage 816) and the village of Wereham, Norfolk (Wigorham 10th C), which incorporates the old name of the Wissey.
These are probably equivalent to Baque ibai-gorri "red river".
In Roman times, there was a town knows as Baekor.../Baegor..., somewhere near Seville. And there was a deity, Baicor(r)ixus, recorded on Roman era inscriptions at Balesta, Huos and Cier de Luchon, all Haute-Garonne. This may have been the tutelary god of the Bigerriones, who could well have derived their name from the old name of the Gave de Pau.
With best wishes,
Angus J Huck